1
$\begingroup$

I would like to find a step by step solutionfor the following Markowitx problem. It is a standard markowitz problem. The unique detail (wich is why I am posting this question here) is that there is a upper bound and a lowerbound for the weights vector.

The problem I want to solve:

enter image description here

The detail:

enter image description here

Definitions:

$w$ is the weight vector, $\Sigma$ is the covariance matrix, and $\mu$ is the returns vector.

| improve this question | |
$\endgroup$
2
$\begingroup$

The constraints $$ w \le b_u $$ and $$ b_l \le w \Leftrightarrow-w \ge - b_l $$ can all be handled using the Kuhn–Tucker conditions. Numerical solvers exist for these linear constraints too (e.g. this is in R). See als this.

However, with the lower bound you often want the optomizer to choose some assets to be zero and if greater zero then greater than some lower bound. In this case you get a mixed-integer problem which are much harder to handle (I have sucessfully applied LINDO to this class of problems).

| improve this answer | |
$\endgroup$
  • $\begingroup$ When you have upper and lower bound constraints there will be additional non-negative Lagrange multipliers $\mu_l\ge 0,\mu_u\ge 0$ (so 2n extra Lagrange mutlipliers) and you will have "complementary slackness" conditions from KKT of the form $\mu_u(b_u-w)=0$. $\endgroup$ – noob2 Mar 22 '17 at 11:55
  • $\begingroup$ @noob2 would you have a derivation (or a reference to one) for this problem? $\endgroup$ – John Mar 22 '17 at 14:44

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.